Import Question JSON

Current Question (ID: 9703)

Question:

$\text{The pulleys and strings shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium, the angle } \theta \text{ should be:}$

Options:

1. $0^\circ$
2. $30^\circ$
3. $45^\circ$
4. $60^\circ$

Solution:

\text{Hint: Balance the horizontal and vertical forces.} \text{Step: Find the value of } \theta \text{From the figure, for the equilibrium of the system, we analyze} \text{the forces on the central mass } \sqrt{2}m\text{.} \text{The tension in the string connected to each mass } m \text{ is } T = mg\text{.} \text{For equilibrium of the central mass } \sqrt{2}m\text{, the vertical} \text{component of tensions must balance its weight:} 2T\cos\theta = \sqrt{2}mg \text{Since } T = mg\text{, we have:} 2mg\cos\theta = \sqrt{2}mg \Rightarrow \cos\theta = \frac{1}{\sqrt{2}} \Rightarrow \theta = 45° \text{Therefore, the value of } \theta \text{ is } 45°\text{.} \text{Hence, option (3) is the correct answer.}

Import JSON File

Upload a JSON file containing LaTeX/MathJax formatted question, options, and solution.

Expected JSON Format:

{
  "question": "The mass of carbon present in 0.5 mole of $\\mathrm{K}_4[\\mathrm{Fe(CN)}_6]$ is:",
  "options": [
    {
      "id": 1,
      "text": "1.8 g"
    },
    {
      "id": 2,
      "text": "18 g"
    },
    {
      "id": 3,
      "text": "3.6 g"
    },
    {
      "id": 4,
      "text": "36 g"
    }
  ],
  "solution": "\\begin{align}\n&\\text{Hint: Mole concept}\\\\\n&1 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\text{ moles of carbon atom}\\\\\n&0.5 \\text{ mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6] = 6 \\times 0.5 \\text{ mol} = 3 \\text{ mol}\\\\\n&1 \\text{ mol of carbon} = 12 \\text{ g}\\\\\n&3 \\text{ mol carbon} = 12 \\times 3 = 36 \\text{ g}\\\\\n&\\text{Hence, 36 g mass of carbon present in 0.5 mole of } \\mathrm{K}_4[\\mathrm{Fe(CN)}_6].\n\\end{align}",
  "correct_answer": 4
}